It is worth keeping in mind that the broader effects might still be positive. But, I think the original poster was just asking about the direct cost of determining who gets the free lunches (that’s why they frame it in terms of the overhead).
While I agree that a full analyst of the social benefit would be better, and I bet it would almost certainly end up being a net positive (and also, the possibility that kids are just not getting fed because of record-keeping screw-ups, missed paperwork, or incomplete programs is just unconscionable), the question the asked about the overhead does have the benefit of being a lot more answerable and direct.
While I agree that a full analyst of the social benefit would be better, and I bet it would almost certainly end up being a net positive (and also, the possibility that kids are just not getting fed because of record-keeping screw-ups, missed paperwork, or incomplete programs is just unconscionable), the question the asked about the overhead does have the benefit of being a lot more answerable and direct.